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Circuit lower bounds for MerlinArthur classes
 In Proc. ACM STOC
, 2007
"... We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ ..."
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Cited by 12 (2 self)
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We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the averagecase setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudorandom generators computable using O(1) bits of advice. 1
Some results on averagecase hardness within the polynomial hierarchy
 In Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science
, 2006
"... Abstract. We prove several results about the averagecase complexity of problems in the Polynomial Hierarchy (PH). We give a connection among averagecase, worstcase, and nonuniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worstcase then it is either ..."
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Cited by 2 (0 self)
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Abstract. We prove several results about the averagecase complexity of problems in the Polynomial Hierarchy (PH). We give a connection among averagecase, worstcase, and nonuniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worstcase then it is either hard on the average (in the sense of Levin) or it is nonuniformly hard (i.e. it does not have small circuits). Recently, Gutfreund, Shaltiel and TaShma (IEEE Conference on Computational Complexity, 2005) showed an interesting worstcase to averagecase connection for languages in NP, under a notion of averagecase hardness defined using uniform adversaries. We show that extending their connection to hardness against quasipolynomial time would imply that NEXP doesn’t have polynomialsize circuits. Finally we prove an unconditional averagecase hardness result. We show that for each k, there is an explicit language in P Σ2 which is hard on average for circuits of size n k. 1
Relativized Collapsing Results under Stringent Oracle Access
"... For relativized arguments, we propose to restrict oracle queries to "stringent" ones. For example, when comparing the power of two machine models relative to some oracle set X, we restrict that machines of both types ask queries from the same segment of the set X. In particular, for investigating po ..."
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For relativized arguments, we propose to restrict oracle queries to "stringent" ones. For example, when comparing the power of two machine models relative to some oracle set X, we restrict that machines of both types ask queries from the same segment of the set X. In particular, for investigating polynomialtime (or polynomialsize) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show, for example, an oracle G such that P , for any constant d 1.
Technical Reports on Mathematical and Computing Sciences: TRC168
"... For relativized arguments, we propose to restrict oracle queries to "stringent" ones; for comparing the power of two machine models relative to some oracle set, stringent relativization is to restrict machines of both types to ask queries on the same segment of the oracle. In particular, for investi ..."
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For relativized arguments, we propose to restrict oracle queries to "stringent" ones; for comparing the power of two machine models relative to some oracle set, stringent relativization is to restrict machines of both types to ask queries on the same segment of the oracle. In particular, for investigating polynomialtime (or polynomialsize) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show an oracle G such that BPP .
Research Reports on
, 2005
"... Employing the variational approach having the twobody reduced density matrix (RDM) as variables to compute the ground state energies of atomicmolecular systems has been a long time dream in electronic structure theory in chemical physics/physical chemistry. Realization of the RDM approach has b ..."
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Employing the variational approach having the twobody reduced density matrix (RDM) as variables to compute the ground state energies of atomicmolecular systems has been a long time dream in electronic structure theory in chemical physics/physical chemistry. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large.
Lower Bounds on Interactive Compressibility by ConstantDepth Circuits
, 2012
"... We formulate a new connection between instance compressibility [HN10]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multiround instance compression. We show that there is no pr ..."
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We formulate a new connection between instance compressibility [HN10]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multiround instance compression. We show that there is no probabilistic multiround compression protocol for Parity in which the computationally bounded party uses a nonuniform AC 0circuit and transmits at most n/(log(n)) ω(1) bits. This result is tight, and strengthens results of Dubrov and Ishai [DI06]. We also show that a similar lower bound holds for Majority. We also consider the question of round separation, i.e., whether for each r � 1, there are functions which can be compressed better with r rounds of compression than with r − 1 rounds. We answer this question affirmatively for compression using constantdepth polynomialsize circuits. Finally, we prove the first nontrivial lower bounds for 1round compressibility of Parity by polynomial size ACC 0 [p] circuits where p is an odd prime.